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On classical solvability for the Hele-Shaw moving boundary problems with kinetic undercooling regularization

Published online by Cambridge University Press:  26 September 2008

Yu. E. Hohlov  and
M. Reissig
Yu. E. Hohlov
Affiliation:
Steklov Mathematical Institute, Vavilova Str. 42, Moscow 117966, Russia
M. Reissig
Affiliation:
Fachbereich Mathematik Technische Universität, Bergakademie Freiberg, Bernhard von Cotta Str. 2, 09596 Freiberg, Germany
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Abstract

In this paper, Hele-Shaw moving boundary problems with kinetic undercooling regularization are studied. By application of the nonlinear abstract Cauchy–Kovalevskaya theorem the local existence of analytic solutions is shown. Therefore we have to study the behaviour of the gradient of harmonic functions up to the boundary of the domain.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 5 , October 1995 , pp. 421 - 439
Copyright
Copyright © Cambridge University Press 1995

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References

[1]Begehr, H. & Gilbert, R. P. 1986 Hele-Shaw type flows in Rn. Nonlinear Analysis 10, 65–85.Google Scholar
[2]Coriel, S. R. & Sekerka, R. F. 1983 Oscillatory morphological instabilities due to nonequilibrium segregation. J. Crystal Growth 61, 499–508.CrossRefGoogle Scholar
[3]Dewynne, J. N., Howison, S. D., Ockendon, J. R. & Xie, W. 1989 Asymptotic behavior of solutions to the Stefan problem with a kinetic condition at the free boundary. J. Austral. Math. Soc. Ser. B 31, 81–96.CrossRefGoogle Scholar
[4]Galin, L. A. 1945 Unsteady filtration with a free surface. Dokl. Akad. Nauk SSSR 47, 250–253 (in Russian).Google Scholar
[5]Gustafsson, B. 1985 Applications of variational inequalities to a moving boundary problem for Hele-Shaw flows. SIAM J. Math. Analysis 16, 279–300.Google Scholar
[6]Hohlov, Yu. E., Howison, S. D., Huntingford, C., Ockendon, J. R. & Lacey, A. A. 1994 A model for non-smooth free boundaries in Hele-Shaw flows. Quart. J. Mech. Math. (to appear).Google Scholar
[7]Hohlov, Y. E. & Howison, S. D. 1994 Ill-Posed Moving Boundary Problems. (Mathematical Monographs Series) Oxford University Press.Google Scholar
[8]Hörmander, L. 1976 The boundary problems of physical geodesy. Arch. Rat. Mech. Anal. 62, 1–52.Google Scholar
[9]Howison, S. D., Lacey, A. A. & Ockendon, J. R. 1985 Singularity development in moving boundary problems. Quart. J. Mech. Math. 38, 343–360.Google Scholar
[10]Howison, S. D. & Xie, W. 1989 Kinetic undercooling regularization of supercooled Stefan problem. In Mathematical Models for Phase Change Problems (Rodrigues, J.-F, ed.). International series of Numerical Mathematics, 88, 227–237, Birkhäuser, Basel.Google Scholar
[11]Howison, S. D. 1992 Complex variable methods in Hele-Shaw moving boundary problems. Euro. J. Appl. Math. 3, 209–224.Google Scholar
[12]Howison, S. D. & Hohlov, Y. E. 1992 O klassificatsii reshenii v zadache o techeniyakh Hele-Shaw s neizvestnoi granitsei. Dokl. Rossiiskoi academii nauk 325, 1161–1166.Google Scholar
[13]May, I. 1994 Untersuchungen zur Existenz und Eindeutigkeit von analytischen Lösungen für Hele-Shaw Strömungen mit kompakten Quellen und Senken. Diplomarbeit, Technischen Universität, Bergakademie Freiberg.Google Scholar
[14]Meirmanov, A. M. 1992 The Stefan Problem, de Gruyter.Google Scholar
[15]Nishida, T. 1977 A note on a theorem of Nirenberg. J. Diff. Geom. 12, 629–633.Google Scholar
[16]Ovsjannikov, L. V. 1965 A singular operator in a scale of Banach spaces. Dokl. Akad. Nauk SSSR 163, 819–822 (in Russian).Google Scholar
[17]Polubarinova-Kochina, P. Ya. 1945 On the motion of the oil contour. Dokl. Akad. Nauk SSSR 47, 254–257 (in Russian).Google Scholar
[18]Reissig, M. & von Wolfersdorf, L. 1993 A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane. Arkiv för matematik 31, 101–116.Google Scholar
[19]Reissig, M. 1994 About the existence and uniqueness of analytic solutions for a moving boundary problem for Hele-Shaw flows in the plane. Nonlinear Analysis 23, 565–576.Google Scholar
[20]Romero, L. A. 1981 The Fingering Problem in a Hele-Shaw Cell. PhD thesis, California Institute of Technology.Google Scholar
[21]Saffman, P. G. 1986 Viscous fingering in Hele-Shaw cells. J. Fluid Mech. 173, 73–94.Google Scholar
[22]Schaefer, R. J. & Glicksman, M. E. 1969 Fully time-dependent theory for the growth of spherical crystal nuclei. J. Crystal Growth 5, 44–58.Google Scholar
[23]Tanveer, S. 1993 Evolution of Hele-Shaw interface for small surface tension. Phil.Trans. R. Soc. Lond. A 343, 155–204.Google Scholar
[24]Tutschke, W. 1978 Vorlesungen über Partielle Differentialgleichungen. Teubner-Texte zur Mathematik. Leipzig.Google Scholar
[25]Vinogradov, Yu. P. & Kufarev, P. P. 1948 On a filtration problem. Prikl. Mat. Mech. 12 (in Russian; English translation as University of Delaware Applied Mathematics Institute Technical Report no. 182A, 1984.)Google Scholar
[26]Xie, W. 1990 The Stefan problem with a kinetic condition at the free boundary. SIAM J. Math. Anal. 21, 362–373.Google Scholar